Adaptive High Order Sliding Mode Controller Design for Hypersonic Vehicle with Flexible Body Dynamics

Abstract

This paper describes the design of a nonlinear robust adaptive controller for a flexible hypersonic vehicle model which is nonlinear, multivariable, and unstable, and includes uncertain parameters. Firstly, a control-oriented model is derived for controller design. Then, the model analysis is conducted for this model via input-output (I/O) linearized technique. Secondly, the sliding mode manifold is designed based on the homogeneity theory. Then, the adaptive high order sliding mode controller is designed to achieve the tracking for hypersonic vehicle where the upper bounds of the uncertainties are not known in advance. Furthermore, the stability of the system is proved via the Lyapunov theory. Finally, the Monte-Carlo simulation results on the full-order nonlinear model with aerodynamic uncertainties are provided to demonstrate the effectiveness of the proposed control strategy.

1. Introduction

The need for a reliable and cost-effective access to space for both civilian and military applications has spurred a renewed interest in hypersonic vehicle. A considerable effort has been made by the US Air Force and NASA to further their development during the last decades. Despite the recent success of NASA’s X-43A experimental vehicle, the control system design for flexible hypersonic vehicle is a challenging task due to the high complexity of equations of motion which stems from strong couplings during propulsive and aerodynamic effects, significant flexibility, the slender geometries and so on. In addition, the hypersonic vehicle is very sensitive to the changes in flight condition and model uncertainties which make the flight control system design more difficult. During the last decades, a kind of flexible hypersonic vehicle model including flexible dynamics has been developed in [1–3]. Based on this model, there have been several papers discussing the challenges associated with the dynamics and control of hypersonic vehicle [4, 5] and many linear and nonlinear control methods have been employed on the flight control system.

Considering the complexity of flexible hypersonic vehicle model, the linear control strategy is applied in flight control system. such As, Groves used Linear Quadratic Regulator (LQR) technique [6], while Hughes applied 𝐻_∞ as well as Linear Parameter Varying (LPV) control method [7], and Serrani used implicit model-following control methods [8–10] to design linear controller for a linearized flexible hypersonic vehicle model at a specified trim condition. Then the linear controller is applied to the nonlinear plant. Although this control strategy performed well on an idealized situation, it is only applied to a single linearized equilibrium position. If a wider operating range is desired, the gain scheduling technique will be required. Using the method, the flight control design is carried out by linearizing the system at a series of operating points and designing separate controllers at each of these points. Finally, the overall flight control system is realized in the philosophy of gain scheduling where the individual gains are interpolated online with respect to some meaningful system parameters such as dynamic pressure and Mach number. However, the gain scheduling has a distinct drawback for the hypersonic vehicle application: the number of required gains to be designed and scheduled within the controller is very large. If one also imposes the design constraint that these gains must allow for a range of possible missions, payloads, and anticipated failure modes, then the number of required gains can become prohibitive [11]. In addition, the method involves the lack of guaranteed global robustness, performance, and especially stability [12].

In order to improve the control performance, several nonlinear control methods are applied in hypersonic vehicle. A nonlinear control method combining an inner loop feedback linearization and an outer loop LQR controller with integral augmentation was proposed by Parker et al. [13]. Although it shows some robustness with respect to small parameter variations, the method neglects the altitude information which may not be a valid assumption in practice. Another nonlinear control method model reference control strategy was presented by Wilcox et al. [14] for hypersonic vehicle with additive bounded disturbances and flexible effects. The temperature-dependent state-space model was proposed to describe the flexible effects. Then a Lyapunov-based continuous robust controller is designed to achieve exponential tracking of a reference model. A potential drawback of the method is that the boundary of parameter uncertainties has to be known exactly in advance which may be difficult to obtain in actual flight. The approaches in [15, 16] decomposed the flight system into several subsystems, where adaptive control and a robust semiglobal gain assignment method has been utilized at each step of the design. In [17, 18], a robust nonlinear tracking controller was constructed by using a mini-max LQR control approach for the flexible hypersonic vehicle. The mini-max optimal control design method is an extension of LQR to a class of uncertain systems. This controller provides robust stability and good performance for the systems under varying flight conditions. In [19], the L1 adaptive control architecture is proposed for hypersonic vehicle with flexible body dynamics where a low-pass filter is used in the feedback loop to guarantee the transient performance for system’s input and output signals. In addition, the flight control problem with actuator delay and uncertainty was investigated in [20] via linear matrix inequality technique. At the same time, robust inversion-based design [21] and slidingmode control [22–24] technology have been proposed for rigid-body hypersonic vehicle models. It is important to note that the controller proposed by Li et al. [23, 24] can achieve the tracking for the response to velocity and altitude in a finite time. However, this vehicle model did not take into account the effect of flexible dynamics and the coupling between thrust and pitching moment. In addition, in order to guarantee the stability of the system, the upper bound of uncertainties has to be exactly known in advance which is difficult to obtain in practice. Therefore, it is necessary to investigate new nonlinear robust adaptive control strategy which avoids the issues mentioned above for flexible hypersonic vehicle.

The motivation of the research is to develop practical nonlinear robust control methods for flexible hypersonic vehicle where the bounds of uncertainties are not known exactly in advance. Furthermore, the main contributions of this paper can be summarized: the control strategy combining input-output linearization and high order sliding mode is proposed for flexible hypersonic vehicle. The key point of that is to consider that the bounds of aerodynamic uncertainties are not known. Then, the adaptive law is designed to adjust the control gains dynamically to ensure the establishment of sliding mode motion. In addition, the characteristic of high order sliding mode is achieved via homogeneity theory as long as the sliding mode motion occurs. The rest of the paper is organized as follows. The problem formulation including vehicle model and control objective is presented in Section 2. In Section 3, the control-oriented model is established for controller design. Then, the adaptive high order sliding mode controller is designed to track the responses to a step change in altitude and airspeed despite the model parameter uncertainties in Section 4. The normal and Monte Carlo tests are conducted for trimmed cruise conditions in altitude 85000 ft and velocity 7710 ft/s in Section 5, whereas the summary and discussion are presented in Section 6.

2. Problem Formulation

2.1. Vehicle Model

The original flexible hypersonic vehicle model considered in the research is the first-principle model (FPM) developed by AFRL/RBCA [25–27]. The equations of motion for the longitudinal model are derived via Lagrange’s equations and compressible flow theory. The flexible modes are considered as a single flexible structure. For convenience, the curve-fitted model (CFM) is proposed by Bolender and Serrani [1–3] via optimization technique. The nonlinear model for the longitudinal dynamics of the CFM consists of five rigid body and six flexible states which can be expressed as 𝑉̇=𝑇cos𝛼−𝐷𝑚−𝑔sin𝛾,𝛾̇=𝐿+𝑇sin𝛼−𝑚𝑉𝑔cos𝛾𝑉,ℎ̇=𝑉sin𝛾,𝛼̇=𝑄−𝛾̇,̇𝑀𝑄=^𝑦𝑦𝐼_𝑦𝑦,𝜂̈_𝑖=−2𝜍_𝑚𝜔_𝑚,𝑖𝜂̇_𝑖−𝜔²_𝑚,𝑖𝜂_𝑖+𝑁_𝑖,𝑖=1,2,3. (1)(2)(3)(4)(5)(6) The five rigid-body states are velocity (𝑉), flight path angle (𝛾), altitude (ℎ), angle of attack (𝛼), and pitch rate (𝑄). The six flexible states are represented as 𝜂_𝑖 and 𝜂̇_𝑖(𝑖=1,2,and3). The approximations lift force 𝐿, drag force 𝐷, thrust 𝑇, pitch moment 𝑀_𝑦𝑦, and generalized forces 𝑁_𝑖(𝑖=1,2,and3) are calculated as follows: 𝐿≈𝑞𝑆𝐶_𝐿𝛼,𝛿_𝑒,𝛿_𝑐,Δ𝜏₁,Δ𝜏₂,𝐷≈𝑞𝑆𝐶_𝐷𝛼,𝛿_𝑒,𝛿_𝑐,Δ𝜏₁,Δ𝜏₂,𝑀_𝑦𝑦≈𝑧_𝑇𝑇+𝑞𝑆𝑐𝐶_𝑀𝛼,Δ𝜏₁,Δ𝜏₂,𝑀_𝑎,𝐴_𝑑,𝑇≈𝑞𝜙𝐶_𝑇,𝜙𝛼,Δ𝜏₁,𝑀_𝑎+𝐶_𝑇𝛼,Δ𝜏₁,𝑀_𝑎,𝐴_𝑑,𝑁_𝑖≈𝑞𝐶_𝑁𝑖𝛼,𝛿_𝑒,𝛿_𝑐,Δ𝜏₁,Δ𝜏₂,𝑖=1,2,3, (7) where 𝐴_𝑑 denotes diffuser-area ratio which is set to constant 1 in the research, 𝑞 is dynamic pressure, and parameters 𝑆,𝑐 are vehicle constants. The aerodynamic forces and thrust coefficients are given by [2] 𝐶_𝐿=𝐶^𝛼_𝐿𝛼+𝐶^𝛿𝑒_𝐿𝛿_𝑒+𝐶^𝛿𝑐_𝐿𝛿_𝑐+𝐶^Δ𝜏1_𝐿Δ𝜏₁+𝐶^Δ𝜏2_𝐿Δ𝜏₂+𝐶⁰_𝐿,𝐶_𝑀=𝐶^𝛼_𝑀𝛼+𝐶^𝛿𝑒_𝑀𝛿_𝑒+𝐶^𝛿𝑐_𝑀𝛿_𝑐+𝐶^Δ𝜏1_𝑀Δ𝜏₁+𝐶^Δ𝜏2_𝑀Δ𝜏₂+𝐶⁰_𝑀,𝐶_𝐷=𝐶^{(𝛼+Δ𝜏1)2}_𝐷𝛼+Δ𝜏₁²+𝐶^{(𝛼+Δ𝜏1)}_𝐷𝛼+Δ𝜏₁+𝐶^𝛿2𝑒_𝐷𝛿²_𝑒+𝐶^𝛿𝑒_𝐷𝛿_𝑒+𝐶^𝛿2𝑐_𝐷𝛿²_𝑐+𝐶^𝛿𝑐_𝐷𝛿_𝑐+𝐶^𝛼𝛿𝑒_𝐷𝛼𝛿_𝑒+𝐶^𝛼𝛿𝑐_𝐷𝛼𝛿_𝑐+𝐶^Δ𝜏2_𝐷Δ𝜏₂+𝐶⁰_𝐷,𝐶_𝑇,𝜙=𝐶^𝛼_𝑇,𝜙𝛼+𝐶^{𝛼𝑀∞−2}_𝑇,𝜙𝛼𝑀_𝑎⁻²+𝐶^𝛼Δ𝜏1_𝑇,𝜙𝛼Δ𝜏₁+𝐶^𝑀∞−2_𝑇,𝜙𝑀_𝑎⁻²+𝐶^Δ𝜏12_𝑇,𝜙Δ𝜏₁²+𝐶^Δ𝜏1_𝑇,𝜙Δ𝜏₁+𝐶⁰_𝑇,𝜙,𝐶_𝑇=𝐶^𝐴𝑑_𝑇𝐴_𝑑+𝐶^𝛼_𝑇𝛼+𝐶^𝑀∞−2_𝑇𝑀_𝑎⁻²+𝐶^Δ𝜏1_𝑇Δ𝜏₁+𝐶⁰_𝑇,𝐶_𝑁𝑖=𝐶^𝛼_𝑁𝑖𝛼+𝐶^𝛿𝑒_𝑁𝑖𝛿_𝑒+𝐶^𝛿𝑐_𝑁𝑖𝛿_𝑐+𝐶^Δ𝜏1_𝑁𝑖Δ𝜏₁+𝐶^Δ𝜏2_𝑁𝑖Δ𝜏₂+𝐶⁰_𝑁𝑖, (8) where available control inputs are throttle setting 𝜙, elevator deflection 𝛿_𝑒, and canard deflection 𝛿_𝑐. The parameters Δ𝜏₁ and Δ𝜏₂ denote the deflections of the fore-body turn angle and aft-body vertex angle, respectively. The reasonability of using these parameters to descript the flexible effect in the curve-fitting model is provided in [2]. The equations that describe the relationship of dynamic pressure, altitude, velocity, and the free-stream Mach number 𝑀_𝑎 are presented as 1𝑞=2𝜌(ℎ)𝑉²,𝑀_𝑎=𝑉𝑀₀,𝑀₀=𝑘𝑅𝑇(ℎ), (9) where 𝑀₀ is the speed of sound, 𝑘=1.4 is the ratio of specific heats, 𝑅=1716 is the gas constant, 𝑇(ℎ) is the temperature, and 𝜌(ℎ) is the atmospheric density which can be obtained based on the 1976 US standard atmosphere.

2.2. Control Objective

The objective of the research is to design control inputs which make the system output tracks the step response for velocity 𝑉_𝑑 and altitude ℎ_𝑑 equivalently satisfying lim_𝑡→∞𝑉−𝑉_𝑑=0,lim_𝑡→∞ℎ−ℎ_𝑑=0, (10) in the presence of aerodynamic parameter uncertainties Δ𝐶_𝐿≤15%𝐶_𝐿,Δ𝐶_𝐷≤15%𝐶_𝐷,Δ𝐶_𝑀≤15%𝐶_𝑀,Δ𝐶_𝑇≤15%𝐶_𝑇,Δ𝐶_𝑇,𝜙≤15%𝐶_𝑇,𝜙. (11) Note that in order to verify the effectiveness of the proposed control strategy, the aerodynamic parameters uncertainties (11) are added to the original nonlinear equations (8).

3. Input-Output Linearization Theory

3.1. Control-Oriented Modeling

For the purpose of feedback linearization, some simplifications are carried out for the flexible hypersonic vehicle. In the simplification process, we removed the flexible states in view of the fact that the flexible modes were demonstrated to have stable dynamics in [2, 13]. The similar simplification can also be found in [18]. Furthermore, the canard deflection 𝛿_𝑐 is chosen as a function of elevator deflection 𝛿_𝑒 which is used to eliminate the nonminimum phase behavior [13]. The relationship between the canard deflection and elevator deflection can be expressed as 𝑘_𝑒𝑐𝐶=−^𝛿𝑒𝐿𝐶^𝛿𝑐_𝐿. (12) Substituting (12) the aerodynamic coefficients defined in (8) results in 𝐶_𝐿=𝐶^𝛼_𝐿𝛼+𝐶⁰_𝐿+Δ𝐶_𝐿,𝐶_𝐷=𝐶^{(𝛼+Δ𝜏1)2}_𝐷𝛼²+𝐶^{(𝛼+Δ𝜏1)}_𝐷𝛼+𝐶⁰_𝐷+Δ𝐶_𝐷,𝐶_𝑀=𝐶^𝛼_𝑀𝛼+𝐶^𝛿𝑒_𝑀−𝐶^𝛿𝑐_𝑀𝐶^𝛿𝑒_𝐿𝐶^𝛿𝑐_𝐿𝛿_𝑒+𝐶⁰_𝑀+Δ𝐶_𝑀,𝐶_𝑇,𝜙=𝐶^𝛼_𝑇,𝜙𝛼+𝐶^{𝛼𝑀∞−2}_𝑇,𝜙𝛼𝑀_∞⁻²+𝐶^𝑀∞−2_𝑇,𝜙𝑀_∞⁻²+𝐶⁰_𝑇,𝜙+Δ𝐶_𝑇,𝜙,𝐶_𝑇=𝐶^𝐴𝑑_𝑇𝐴_𝑑+𝐶^𝛼_𝑇𝛼+𝐶^𝑀∞−2_𝑇𝑀_∞⁻²+𝐶⁰_𝑇+Δ𝐶_𝑇, (13) where Δ𝐶_𝐿, Δ𝐶_𝐷, Δ𝐶_𝑀, Δ𝐶_𝑇,𝜙, and Δ𝐶_𝑇 are considered as aerodynamic parameter uncertainties and defined as Δ𝐶_𝐿=𝐶^Δ𝜏1_𝐿Δ𝜏₁+𝐶^Δ𝜏2_𝐿Δ𝜏₂,Δ𝐶_𝐷=2𝐶^{(𝛼+Δ𝜏1)2}_𝐷𝛼Δ𝜏₁+𝐶^{(𝛼+Δ𝜏1)2}_𝐷Δ𝜏₁²+𝐶^{(𝛼+Δ𝜏1)}_𝐷Δ𝜏₁+𝐶^𝛿2𝑒_𝐷𝛿²_𝑒+𝐶^𝛿𝑒_𝐷𝛿_𝑒+𝐶^𝛿2𝑐_𝐷𝛿²_𝑐+𝐶^𝛿𝑐_𝐷𝛿_𝑐+𝐶^𝛼𝛿𝑒_𝐷𝛼𝛿_𝑒+𝐶^𝛼𝛿𝑐_𝐷𝛼𝛿_𝑐+𝐶^Δ𝜏2_𝐷Δ𝜏₂,Δ𝐶_𝑀=𝐶^Δ𝜏1_𝑀Δ𝜏₁+𝐶^Δ𝜏2_𝑀Δ𝜏₂,Δ𝐶_𝑇,𝜙=𝐶^𝛼Δ𝜏1_𝑇,𝜙𝛼Δ𝜏₁+𝐶^Δ𝜏12_𝑇,𝜙Δ𝜏₁²+𝐶^Δ𝜏1_𝑇,𝜙Δ𝜏₁,Δ𝐶_𝑇=𝐶^Δ𝜏1_𝑇Δ𝜏₁. (14) Inspired by [22], we dynamically extend the system by introducing second-order actuator dynamics as follows: 𝜙̈=−2𝜍𝜔_𝑛𝜙̇−𝜔²_𝑛𝜙+𝜔²_𝑛𝜙_𝑐. (15) The actuator dynamics are chosen to impose a damping ratio 𝜁=0.7 and a natural frequency 𝜔_𝑛=5rad/s.

Through analysis, the control-oriented model is composed of (1)–(5) with aerodynamic coefficients equations (13)-(14) and actuator dynamics equation (15) which will be considered as the controller design model in the next. It is easily observed that the order of the control-oriented model is equal to 7.

Remark 1. Although the flexible modes have been removed in the process of controller design, the original full-order flexible hypersonic vehicle model consisting of (1)–(8) will be used for simulation verification in the research.

3.2. Input-Output Linearization for Control-Oriented Model

Applying the input-output linearization technique [28] to the control-oriented model, the input-output dynamics can be derived by differentiating velocity three times and differentiating altitude four times as seen below. Therefore, the relative degree of the system equals the order of the system. Hence, the control-oriented model can be linearized completely as follows: 𝑉⃛ℎ⁽⁴⁾=𝐹_𝑉+Δ𝐹_𝑉𝐹_ℎ+Δ𝐹_ℎ+𝐺₁₁+Δ𝐺₁₁𝐺₁₂+Δ𝐺₁₂𝐺₂₁+Δ𝐺₂₁𝐺₂₂+Δ𝐺₂₂𝐮, (16) where the system uncertainty parts Δ𝐹_𝑉, Δ𝐹_ℎ, Δ𝐺₁₁, Δ𝐺₁₂, Δ𝐺₂₁, and Δ𝐺₂₂ are caused by the aerodynamic uncertainties, unmodeled dynamics, coupling between the rigid-body and flexible-body, and so on. The control input used in (16) is defined as 𝐮=[𝑢₁,𝑢₂]^𝑇=[𝜙_𝑐,𝛿_𝑒]^𝑇. The elements of system matrix and control matrix are calculated as follows: 𝐹_𝑉=𝜔₁𝑥̈₀+𝑥̇^𝑇Ω₂𝑥̇𝑚,𝐹_ℎ=3𝑉̈𝛾̇cos𝛾−3𝑉̇𝛾̇²𝑉̇̈𝛾̇̈𝛾̇sin𝛾+3𝛾cos𝛾−3𝑉𝛾sin𝛾−𝑉³𝜔cos𝛾+sin𝛾₁𝑥̈₀+𝑥̇^𝑇Ω₂𝑥̇𝑚+𝑉cos𝛾𝜋₁𝑥̈₀+𝑥̇^𝑇Π₂𝑥̇,𝐺₁₁=1𝑚𝜕𝑇𝜕𝜙cos𝛼𝜔²_𝑛,𝐺₁₂=𝜌𝑉²𝑆𝑐2𝑚𝐼_𝑦𝑦𝐶^𝛿𝑒_𝑀−𝐶^𝛿𝑒_𝐿𝐶^𝛿𝑐_𝑀𝐶^𝛿𝑐_𝐿×𝜕𝑇𝜕𝛼cos𝛼−𝑇sin𝛼−𝜕𝐷,𝐺𝜕𝛼₂₁=1𝑚𝜕𝑇𝜔𝜕𝜙^2𝑛,𝐺sin𝛼+𝛾₂₂=𝜌𝑉²𝑆𝑐2𝑚𝐼_𝑦𝑦𝐶^𝛿𝑒_𝑀−𝐶^𝛿𝑒_𝐿𝐶^𝛿𝑐_𝑀𝐶^𝛿𝑐_𝐿×cos𝛾𝜕𝑇𝜕𝛼sin𝛼+𝑇cos𝛼+𝜕𝐿𝜕𝛼+sin𝛾𝜕𝑇𝜕𝛼⋅cos𝛼−𝑇sin𝛼−𝜕𝐷,𝜕𝛼 (17) where 𝐱^𝑇=[𝑉,𝛾,𝛼,𝜙,ℎ],𝐱̈^𝑇₀=[𝑉̈,𝛾̈,𝛼̈₀,𝜙̈₀,ℎ̈]. For brevity, the specific expression of 𝛼̈₀, 𝜙̈₀, 𝜔₁, 𝜋₁, 𝛀₂, and 𝚷₂ can be obtained in the Appendix.

4. Adaptive High Order Sliding Mode Control Strategy

4.1. Sliding Mode Manifold Design

Inspired by the work provided in [29], the sliding mode manifold for velocity and altitude in system (16) is designed based on homogeneity theory [30] as follows𝑠_𝑉=𝜎̈_𝑉+𝜆_1𝑉𝜎^𝑎1𝑉_𝑉sign𝜎_𝑉+𝜆_2𝑉𝜎̇^𝑎2𝑉_𝑉sign𝜎̇_𝑉+𝜆_3𝑉𝜎̈^𝑎3𝑉_𝑉sign_𝑉𝐶𝑉𝜎̈𝑠𝑑𝑡,_ℎ=𝜎⃛_ℎ+𝜆_1ℎ𝜎^𝑎1ℎ_ℎsign𝜎_ℎ+𝜆_2ℎ𝜎̇^𝑎2ℎ_ℎsign𝜎̇_ℎ+𝜆_3ℎ𝜎̈^𝑎3ℎ_ℎsign𝜎̈_ℎ+𝜆_4ℎ𝜎⃛^𝑎4ℎ_ℎsign_ℎ𝐶ℎ𝜎⃛𝑑𝑡, (18)

where 𝜎_𝑉=𝑉−𝑉_𝑑,𝜎_ℎ=ℎ−ℎ_𝑑. The parameters 𝜆_𝑖𝑉(𝑖=1,2,and3),𝜆_𝑗ℎ(𝑗=1,2,3,and4) are selected to be positive such that the polynomials 𝜆_3𝑉𝑠²+𝜆_2𝑉𝑠+𝜆_1𝑉 and 𝜆_4ℎ𝑠⁴+𝜆_3ℎ𝑠²+𝜆_2ℎ𝑠+𝜆_1ℎ are Hurwitz. The constants 𝑎_𝑖𝑉(𝑖=1,2,and3),𝑎_𝑗ℎ(𝑗=1,2,3,and4) are calculated as 𝑎_{(𝑖−1)𝑉}=𝑎_𝑖𝑉𝑎_(𝑖+1)𝑉2𝑎_(𝑖+1)𝑉−𝑎_𝑖𝑉{},𝑎,𝑖∈2,3_{(𝑗−1)ℎ}=𝑎_𝑗ℎ𝑎_(𝑗+1)ℎ2𝑎_(𝑗+1)ℎ−𝑎_𝑗ℎ,𝑗∈{2,3,4}, (19) with 𝑎_4𝑉=𝑎_5ℎ=1 and 𝑎_3𝑉∈(1−𝜀_𝑉,1),𝑎_4ℎ∈(1−𝜀_ℎ,1), where 𝜀_𝑉∈(0,1),𝜀_ℎ∈(0,1).

Based on the definition in (18), it can be observed that if 𝑠_𝑉=0,𝑠_ℎ=0 is achieved, then the following equations are satisfied: 𝜎⃛_𝑉=−𝜆_1𝑉𝜎^𝑎1𝑉_𝑉sign𝜎_𝑉+𝜆_2𝑉𝜎̇^𝑎2𝑉_𝑉sign𝜎̇_𝑉+𝜆_3𝑉𝜎̈^𝑎3𝑉_𝑉sign𝜎̈_𝑉,𝜎_ℎ⁽⁴⁾=−𝜆_1ℎ𝜎^𝑎1ℎ_ℎsign𝜎_ℎ+𝜆_2ℎ𝜎̇^𝑎2ℎ_ℎsign𝜎̇_ℎ+𝜆_3ℎ𝜎̈^𝑎3ℎ_ℎsign𝜎̈_ℎ+𝜆_4ℎ𝜎⃛^𝑎4ℎ_ℎsign𝜎⃛_ℎ. (20) In view of the homogeneity theory [30], if (20) are established then 𝜎_𝑉,𝜎̇_𝑉, and 𝜎̈_𝑉, and 𝜎_ℎ,𝜎̇_ℎ,𝜎̈_ℎ, and𝜎⃛_ℎ converge to zero finally. Therefore, the objective of the research is equivalent to designing controller which makes the sliding mode motion for (18) occur.

Remark 2. The derivatives for 𝜎_𝑉, 𝜎_ℎ in (18) are calculated on-line via arbitrary order robust sliding mode differentiator as follows.

Lemma 3 (see [31]). Let the signal 𝑠_𝑗(𝑗=𝑉,ℎ) be a function consisting of a bounded Lebesgue-measurable noise with unknown features. There exists a known Lipschitz constant 𝐿_𝑗 such that |𝑠_𝑗^(𝑛+1)|≤𝐿_𝑗. To find real-time robust estimations for 𝑠_𝑗,𝑠_𝑗̇,…,𝑠_𝑗^(𝑛), the following formulation is used: 𝑧̇_0𝑗=𝑣_0𝑗,𝑣_0𝑗=−𝜆₀𝐿_𝑗^1/(𝑛+1)𝑧_0𝑗−𝑠_𝑗^{𝑛/(𝑛+1)}sign𝑧_0𝑗−𝑠_𝑗+𝑧_1𝑗,𝑧̇_1𝑗=𝑣_1𝑗,𝑣_1𝑗=−𝜆₁𝐿_𝑗^1/𝑛𝑧_1𝑗−𝑣_0𝑗^{(𝑛−1)/𝑛}sign𝑧_1𝑗−𝑣_0𝑗+𝑧_2𝑗⋮𝑧̇_{(𝑛−1)𝑗}=𝑣_{(𝑛−1)𝑗},𝑣_{(𝑛−1)𝑗}=−𝜆_𝑛−1𝐿_𝑗^1/2𝑧_{(𝑛−1)𝑗}−𝑣_{(𝑛−2)𝑗}^1/2×sign𝑧_{(𝑛−1)𝑗}−𝑣_{(𝑛−2)𝑗}+𝑧_𝑛𝑗,𝑧̇_𝑛𝑗=−𝜆_𝑛𝐿_𝑗sign𝑧_𝑛𝑗−𝑣_{(𝑛−1)𝑗}. (21)

If the parameters are properly chosen, the following equalities are true in the absence of input noise after a finite time of a transient process: 𝑧_0𝑗=𝑠_𝑗,𝑧_𝑖𝑗=𝑠_𝑗^(𝑖),𝑖=1,2,…,𝑛;𝑗=𝑉,ℎ. (22)

Actually the third-order differentiator is enough to calculate the derivatives. However, to obtain smooth differential signal, the fifth-order differentiator with parameters 𝐿_𝑉=100,𝐿_ℎ=10,𝜆₀=12,𝜆₁=8,𝜆₂=5,𝜆₃=1.5,𝜆₃=3, and 𝜆₃=1.1 is used to calculate the derivatives in (18) in the research.

4.2. Adaptive High Order Sliding Mode Controller Design

In this section, the adaptive high order sliding mode control strategy is proposed to achieve the tracking for the response of step change for velocity and altitude in the presence of model uncertainties.

Differentiating (18) results in 𝑠_𝑉̇𝑠_ℎ̇=𝐹_𝑉+𝐶_𝑉−𝑉_𝑑⁽³⁾+Δ𝐹_𝑉𝐹_ℎ+𝐶_ℎ−ℎ_𝑑⁽⁴⁾+Δ𝐹_{ℎ𝐅=𝐅𝐧+Δ𝐅}+𝐺₁₁+Δ𝐺₁₁𝐺₁₂+Δ𝐺₁₂𝐺₂₁+Δ𝐺₂₁𝐺₂₂+Δ𝐺_{22𝐆=𝐆𝐧+Δ𝐆}𝐮, (23) where 𝐅_𝐧=[𝐹_𝑉+𝐶_𝑉−𝑉_𝑑⁽³⁾,𝐹_ℎ+𝐶_ℎ−ℎ_𝑑⁽⁴⁾]^𝑇 and 𝐆_𝐧=[𝐺₁₁,𝐺₁₂;𝐺₂₁,𝐺₂₂] are the nominal parts of the system. The vector Δ𝐅=[Δ𝐹_𝑉,Δ𝐹_ℎ]^𝑇 and matrix Δ𝐆=[Δ𝐺₁₁,Δ𝐺₁₂;Δ𝐺₂₁,Δ𝐺₂₂] are the uncertainty parts.

Assumption 1. The normal matrix 𝐆_𝐧 is nonsingular. In addition, we assume that there are constants 𝜅₁ and 0≤𝜅₂<1 such that the uncertainties discussed in the research satisfy the following condition: Δ𝐅−Δ𝐆𝐆_𝐧⁻¹𝐅_𝐧≤𝜅₁,Δ𝐆𝐆_𝐧⁻¹≤𝜅₂, (24) where the finite boundaries for 𝜅₁ and 𝜅₂ exist but are not known in advance.

The main results can be summarized as the following theorem.

Theorem 4. Considering system (16) with the uncertainties satisfying (24), if the controller is designed as (25) and update law (26), the trajectory of the closed-loop system can be driven onto the sliding mode surface 𝑠_𝑉=𝑠_ℎ=0 without need of the information consider, for the boundaries of 𝜅₁ and 𝜅₂ in advance. 𝐮=𝐆_𝐧⁻¹−𝐅_𝐧−𝜅sign(𝐬)−𝜂sign(𝐬), (25) with adaptive law 𝜅̇=1𝑝‖𝐬‖, (26) where 𝐬=[𝑠_𝑉,𝑠_ℎ]^𝑇,sign(𝐬)=[sign(𝑠_𝑉)sign(𝑠_ℎ)]^𝑇, 𝜅(0)>0𝑝>0, and 𝜂 is an arbitrary small positive constant.

Proof. The Lyapunov candidate function is constructed as 1𝑉=2𝐬^𝑇𝐬+𝑝1−𝜅₂𝜅𝜅−¹1−𝜅₂². (27) According to Assumption 1, it is observed that (27) is positive definite. The derivative of the Lyapunov function candidate is presented 𝑉̇=𝐬^𝑇𝐬̇+𝑝1−𝜅₂𝜅−𝜅₁1−𝜅₂𝜅̇. (28) Substituting (23), and (25)-(26) in (28) yields 𝑉̇=𝐬^𝑇𝐈+Δ𝐆𝐆_𝐧⁻¹−(𝐬)(𝐬)+𝜅sign−𝜂signΔ𝐅−Δ𝐆𝐆_𝐧⁻¹𝐅_𝐧+1−𝜅₂𝜅𝜅−¹1−𝜅₂𝐬‖‖𝐬‖+‖≤−𝜅+𝜂Δ𝐆𝐆_𝐧⁻¹‖𝐬‖+𝜅+𝜂Δ𝐅−Δ𝐆𝐆_𝐧⁻¹𝐅_𝐧‖𝐬‖+1−𝜅₂𝜅𝜅−¹1−𝜅₂‖𝐬‖≤−1−𝜅₂𝜅+𝜂‖𝐬‖+𝜅₁‖𝐬‖+1−𝜅₂𝜅−𝜅₁‖𝐬‖=−𝜂1−𝜅₂‖𝐬‖. (29) In view of Assumption 1, it is obviously that the derivative of Lyapunov function with respect to time is negative, that is, 𝑉̇<0. Thus, the closed-system is asymptotically stable that is, lim_𝑡→∞𝑠_𝑉=0, and lim_𝑡→∞𝑠_ℎ=0. Furthermore, the sliding mode motion will be kept which means the system states evolve on the sliding mode manifold in spite of uncertainties which can be proven via the proposed theorem. Then, according to the definition of the sliding mode surface in (18), it can be concluded that 𝜎_𝑉,𝜎̇_𝑉, and 𝜎̈_𝑉 and 𝜎_ℎ,𝜎̇_ℎ,𝜎̈_ℎ, and 𝜎⃛_ℎ converge to zero as 𝑡→∞ which means our objective is achieved.

Implement Issues. (a) In practice, |𝑠_𝑖|(𝑖=𝑉,ℎ) cannot be zero exactly. Thus the adaptive gain will increase boundlessly. To avoid this drawback, the following technique is used: 𝜅̇=1𝑝‖𝐬‖,|𝐬|̇|𝐬|≥𝜀,𝜅=0,<𝜀, (30) where 𝜀 is an arbitrary small positive constant.

(b) In order to reduce the control chattering, the saturation function with boundary layer 0.01 is used to replace the sign function given in (25) in the simulation.

Remark 5. The main idea of designing the adaptive law in (30) is to dynamically increase the control gain 𝜅 until condition |𝐬|≤𝜀 is satisfied. As soon as the domain is reached, the gain 𝜅 stops increasing and keeps constant. The above process will be repeated, when the sliding mode |𝐬| deviates from the domain 𝜀. It is obvious that the gain 𝜅 increases as 𝜀 decreases provided that the other conditions are fixed. Furthermore, the large control gain leads to the higher accuracy and the faster adjustment ability at the cost of control chattering or the larger control energy. Hence, the choice of parameter 𝜀 is a compromise between the tracking accuracy and control performance.

Remark 6. The main differences between our research and adaptive sliding mode control provided by Xu et al. [22] can be summarized: The proposed adaptive high order sliding mode (HOSM) control method can be extended to arbitrary order sliding mode control. Hence, it has the HOSM’s advantages comparing with the traditional sliding mode control, such as the higher accuracy and the reduced control chattering [31]. The adaptive method discussed in [22] is based on the assumption that the control matrix can be written as a product of a fixed basis matrix and an uncertainty parameter matrix. However, the assumption may not be satisfied for most control systems such as the problem discussed in our research. Furthermore, the adaptive method proposed by Xu is not used to estimate the control gain but the uncertainty parameters. Hence, the guess for control gain has to be made in the simulation. However, our proposed method avoids the repeated guess via designing an adaptive law to estimate the control gain.

5. Simulation Results

5.1. Simulation Parameters Setting

The aerodynamic reference area 𝑆=17ft², mean aerodynamic chord 𝑐=17ft, and moment of inertia 𝐼_𝑦𝑦=500000slug⋅ft²/(rad⋅ft). The aerodynamic coefficients used in the research are the same as that provided in [2]. For brevity, the details are omitted. The simulation is carried out for the trimmed cruise conditions in altitude 85000 ft and velocity 7710 ft/s. The controller parameters used in the simulation are provided in Table 1.

Table 1 

Controller parameters used in the simulation.

5.2. Simulation Results Analysis

5.2.1. Case A: Normal Test

In order to verify the performance of the controller, the simulations have been done on the full nonlinear flexible hypersonic vehicle described in the previous. The simulation results provided in Figures 1 and 2 show the response of the step change for normal model which means no additional aerodynamic uncertainties, that is, (11), are included. In Figures 1(a) and 1(b), it is observed that the velocity and altitude converge to desired value in a short time with almost no overshoot. Figures 1(d) and 1(e) show the velocity error, altitude error and their derivatives information. From that, we find that not only the errors but also their derivatives converge to zero which demonstrate the high order sliding mode motion 𝜎_𝑉=𝜎̇_𝑉=𝜎̈_𝑉=0 and 𝜎_ℎ=𝜎̇_ℎ=𝜎̈_ℎ=𝜎⃛_ℎ=0 occurrence. The angle of attack and flight path angle are given in Figures 1(c) and 1(f), respectively, where the curves change smoothly can be seen.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Vehicle states: (a) velocity change, (b) altitude change, (c) angle of attack, (d) velocity error information, (e) altitude error information, and (f) flight path angle.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Vehicle flexible states, and control: (a) the first flexible modes, (b) the second flexible modes, (c) the third flexible modes, (d) adaptive gain, (e) throttle setting, and (f) elevator deflection.

Furthermore, the six flexible states are provided in Figures 2(a)–2(c) which means the flexible states converge to trimmed points after a short time. The simulation results in Figure 2(d) show the adaptive gain obtained using (30). The control variables, throttle setting and elevator deflection, are show in Figures 2(e) and 2(f), respectively. From the simulation results provided in Figures 1 and 2, it is seen that the proposed adaptive high order sliding mode controller provides good tracking for the full nonlinear flexible hypersonic vehicle.

5.2.2. Case B: Monte-Carlo Test

In order to verify the robustness of the controller, the 500 times Monte-Carlo simulation with the uncertainties provided in (11) is carried out in this section. The main states and control information are plotted in Figure 3. In Figures 3(a) and 3(b), it is observed that the velocity and altitude can track the desired commands in spite of the uncertainties. The angle of attack is shown in Figure 3(c). From that, we can see that the angle of attack has different steady-state values due to the existence of the uncertainties defined in (11). For example, the angle of attack has to be increased, when the drag force coefficient has positive uncertainty, and lift force coefficient has negative uncertainty. The similar cases can also be found from Figures 3(e) and 3(f), whereas the steady-state value of flight path angle is not affected by the aerodynamic uncertainties which can be verified from Figure 3(d). Finally, the simulation results shown in Figure 3 demonstrate that the proposed control strategy has good performance and exhibits robustness with respect to aerodynamic uncertainties.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

500 times Monte-Carlo simulation: (a) velocity change, (b) altitude change, (c) angle of attack, (d) flight path angle, (e) throttle setting, and (f) elevator deflection.

6. Conclusions and Future Work

A nonlinear robust control strategy based on the adaptive high order sliding mode theory is proposed for flexible hypersonic vehicle with the upper bounds of the uncertainties not known in advance. The proposed controller achieves the tracking for the response of step change for velocity and altitude despite the aerodynamic parameter uncertainties. Additionally, an adaptive tuning law is designed to guarantee that prior knowledge about the upper of the system uncertainty is no longer a necessary requirement for the traditional high order sliding mode controller design. In addition, the Monte-Carlo simulation is conducted to verify the effectiveness of the proposed control strategy. The nonlinear robust control for flexible hypersonic vehicle with input constraint will be considered in the future work.

Appendix

The detailed expressions of the vectors 𝛼̈₀,𝜙̈₀,𝜔₁, and 𝜋₁ and matrices Ω₂ and Π₂ are calculated as 𝛼̈₀=1𝐼_𝑦𝑦𝑧_𝑇1𝑇+2𝜌𝑉²𝑆𝑐𝐶^𝛼_𝑀𝛼+𝐶⁰_𝑀−̈𝜙̈𝛾,₀=−2𝜍𝜔_𝑛𝜙̇−𝜔_𝑛²𝜔𝜙,_1𝑇=𝜕𝑇𝜕𝑉cos𝛼−𝜕𝐷𝜕𝑉−𝑚𝑔cos𝛾𝜕𝑇𝜕𝛼cos𝛼−𝑇sin𝛼−𝜕𝐷𝜕𝛼𝜕𝑇−𝜕𝜙cos𝛼𝜕𝑔,𝜋𝜕ℎ𝑚sin𝛾_1𝑇=1𝑉𝜕𝐿+𝜕𝑉𝜕𝑇−𝜕𝑉sin𝛼𝑇sin𝛼+𝐿𝑉²+𝑚𝑔cos𝛾𝑉²𝑚𝑔sin𝛾𝑉1𝑉𝜕𝑇𝜕𝛼sin𝛼+𝑇cos𝛼+𝜕𝐿1𝜕𝛼𝑉𝜕𝑇−1𝜕𝜙sin𝛼𝑉𝜕𝑔.𝜕ℎ𝑚cos𝛾 (A.1)

Define Ω₂=[𝜔₂₁𝜔₂₂𝜔₂₃𝜔₂₄𝜔₂₅], where 𝜔₂₁=𝜕²𝑇𝜕𝑉²𝜕cos𝛼−²𝐷𝜕𝑉²0𝜕²𝑇𝜕𝑉𝜕𝛼cos𝛼−𝜕𝑇𝜕𝜕𝑉sin𝛼−²𝐷𝜕𝜕𝑉𝜕𝛼₂𝑇0,𝜔𝜕𝑉𝜕𝜙cos𝛼₂₂=000−𝑚𝑔sin𝛾𝜕𝑔𝜕ℎcos𝛾,𝜔₂₅=0−𝜕𝑔00−𝜕𝜕ℎ𝑚cos𝛾₂𝑔𝜕ℎ²,𝜔𝑚sin𝛾₂₃=𝜕²𝑇𝜕𝛼𝜕𝑉cos𝛼−𝜕𝑇𝜕𝜕𝑉sin𝛼−²𝐷0𝜕𝜕𝛼𝜕𝑉₂𝑇𝜕𝛼²cos𝛼−2sin𝛼𝜕𝑇𝜕𝜕𝛼−𝑇cos𝛼−²𝐷𝜕𝛼²𝜕²𝑇𝜕𝛼𝜕𝜙cos𝛼−𝜕𝑇0,𝜔𝜕𝜙sin𝛼₂₄=𝜕²𝑇0𝜕𝜕𝜙𝜕𝑉cos𝛼₂𝑇𝜕𝜙𝜕𝛼cos𝛼−𝜕𝑇𝜕𝜕𝜙sin𝛼₂𝑇𝜕𝜙²0.cos𝛼 (A.2)

Define Π₂= [𝜋₂₁𝜋₂₂𝜋₂₃𝜋₂₄𝜋₂₅], where 𝜋₂₁=2(𝑇sin𝛼+𝐿)𝑉³−2𝑉²𝜕𝐿+𝜕𝑉𝜕𝑇+1𝜕𝑉sin𝛼𝑉𝜕²𝐿𝜕𝑉²+𝜕²𝑇𝜕𝑉²sin𝛼−2𝑚𝑔cos𝛾𝑉³−𝑚𝑔sin𝛾𝑉²−1𝑉²𝜕𝑇𝜕𝛼sin𝛼+𝑇cos𝛼+𝜕𝐿+1𝜕𝛼𝑉𝜕²𝐿+𝜕𝜕𝑉𝜕𝛼²𝑇𝜕𝑉𝜕𝛼sin𝛼+𝜕𝑇−1𝜕𝑉cos𝛼𝑉²𝜕𝑇1𝜕𝜙sin𝛼+𝑉𝜕²𝑇1𝜕𝑉𝜕𝜙sin𝛼𝑉²𝜕𝑔,𝜋𝜕ℎ𝑚cos𝛾₂₂=−𝑚𝑔sin𝛾𝑉²𝑚𝑔cos𝛾𝑉00𝜕𝑔𝜕ℎ𝑚sin𝛾𝑉,𝜋₂₅=1𝑉²𝜕𝑔1𝜕ℎ𝑚cos𝛾𝑉𝜕𝑔001𝜕ℎ𝑚sin𝛾𝑉𝜕²𝑔𝜕ℎ²,𝜋𝑚cos𝛾₂₃=−1𝑉²𝜕𝑇𝜕𝛼sin𝛼+𝑇cos𝛼+𝜕𝐿+1𝜕𝛼𝑉𝜕²𝐿𝜕𝜕𝛼𝜕𝑉sin𝛼+²𝑇𝜕𝛼𝜕𝑉cos𝛼+𝜕𝑇01𝜕𝑉cos𝛼𝑉𝜕²𝑇𝜕𝛼²sin𝛼+2𝜕𝑇𝜕𝜕𝛼cos𝛼−𝑇sin𝛼+²𝐿𝜕𝛼²1𝑉𝜕²𝑇𝜕𝛼𝜕𝜙sin𝛼+𝜕𝑇0,𝜋𝜕𝜙cos𝛼₂₄=−1𝑉²𝜕𝑇1𝜕𝜙sin𝛼+𝑉𝜕²𝑇01𝜕𝜙𝜕𝑉sin𝛼𝑉𝜕²𝑇𝜕𝜙𝜕𝛼sin𝛼+𝜕𝑇1𝜕𝜙cos𝛼𝑉𝜕²𝑇𝜕𝜙²0.sin𝛼 (A.3)

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (61203012, 61273092, and 91016018), Key Grant Project of Chinese Ministry of Education (311012), and Tianjin Research Program of Application Foundation and Advanced Technology (11JCZDJC25100, 12JCZDJC30300).